In the final part
of Geometry from Africa (1999) I explained how
I discovered Lunda-designs in the context of analysing mathematical aspects
of traditional sona pictograms from Angola and then I presented briefly
some properties and generalisations of Lunda-designs. Figure
1 presents some examples of Lunda-designs. In several earlier papers
(1990,
1997a, 1999b:
http://members.tripod.com/vismath/paulus/)
and book chapters (1993-4, 1995: 379-428,
1997:
275-300), and in the book
Lunda-Geometry: Designs, Polyominoes, Patterns,
Symmetries (1996) I analysed geometrical properties
of Lunda-designs. As Lunda-designs are often aesthetically attractive,
some papers deal with art and Lunda-designs (2002,
2002a).
Another paper (2000) shows how infinitely many magic
squares may be constructed from Lunda-designs. My contribution to the conference
Symmetry
2000 presents further generalisations of the concept of Lunda-design
(2001). Lunda-designs of Celtic knots are analysed in
(1999a).

Figure 1

In one of the chapters
of my forthcoming book The beautiful geometry and linear algebra of
Lunda-designs (2002b) I explain in detail how experimentation
with Liki-designs, a particular class of Lunda-designs, led me to the discovery
of cycle matrices. The present paper gives an overview.

Liki-designs

It was on the eve
of the fourth anniversary of my daughter Likilisa that I started to analyse
a particular class of Lunda-designs. As these designs turned out to have
some interesting properties I gave them the name of Liki-designs.

Figure 2

Figure
2 presents an example of a Liki-design with its associated Liki-matrix.
The matrices are obtained by substituting the unit squares of one colour
(in this case red) by 0s and the unit squares of the other colour (green
in this case) by 1s.

Figure 3

Figure
4

Lunda-designs are
characterised by the following two local (two-colour) symmetry properties:

(i) Along the border each grid point
always has one red unit square and one green unit square associated with
it (see the example in Figure 3);

(ii) Of the four unit squares between
two arbitrary (vertical or horizontal) neighbouring grid points, two are
red and two are green (see the examples in Figure 4).

Figure
5

In the case of Liki-designs,
the second property is substituted by the following stronger condition:

(ii) Of the four unit squares between
two arbitrary (vertical or horizontal) neighbouring grid points, two neighbouring
unit squares are always red, while the other two are green (see Figure
5).

Figure
6

The new condition
(ii) may be described as follows. Consider the four
unit squares between two vertically or horizontally neighbouring grid points.
Two of them that belong to different rows and different columns always
have different colours (Figure 6).

The two properties
(i) and (ii) imply that a square Liki-design
and its associated Liki-matrix are composed of cycles of alternating red
and green unit squares and of cycles of alternating 0s and 1s, respectively.

3×3 Liki-design B

Liki-matrix B

First cycle

Second cycle

Third cycle

Cycle structure

Figure
7

Figure
7 illustrates the case of the 3×3 Liki-design
B. The corresponding
Liki-matrix has dimensions 6×6.

A question that
naturally emerges is what will happen with the powers of Liki-matrices?

B1

B2

B3

B4

Figure 8

First cycle

Second cycle

Third cycle

Figure 9

a

b

c

d

Figure 10

First order cycle structure

Second order cycle structure

Figure 11

Figure
8 displays the first four powers of Liki-matrix
B. The third
power has the same cycle structure as the first power: the first cycle
of the third power is composed of alternating 3s and 6s, the second cycle
of alternating 2s and 7s and the third cycle, once again, of alternating
3s and 6s (Figure 9). The even powers do not have the
same cycle structure. Their diagonals are constant and they present other
cycles, like the cycle of 2s (Figure 10) of the second
power.
Figure 11 compares the cycle structures of the
odd and even powers of the Liki-matrix
B. A cycle structure of the
first type I call a first order cycle
structure. A cycle structure of the
second type I call a second order cycle
structure. The first order cycles
of the odd powers of
B seem to be composed of alternating numbers.
In other words, these cycles have period 2. The second order cycles of
the even powers of B seem to be constant. In other words, the cycles
have period 1.

Instead of considering powers of
a Liki-design, we may consider the multiplication of Liki-designs. The
following two theorems are proven in (2002b):

Theorem 1:

Let A and
B be Liki-matrices of dimensions (2m) × (2m).
AB has a second order cycle structure of period 1.

Theorem 2:

If A and
B are Liki-matrices of dimensions (2m) ×
(2m), then
A and
B commute: AB=BA.

Figure 12 illustrates
the example of the multiplication of two 4×4
Liki-designs.

First level generalisation

Now we may define
square Liki-matrices in another way. Square Liki-matrices are square {0,1}-matrices
of even dimensions that have a first order cycle structure; each cycle
has period 2. With this redefinition, we are immediately invited to consider
a first level of generalisation of Liki-matrices and their powers: matrices
of even dimensions that have a first or a second order cycle structure
of period 2.

Matrix

First order cycle structure

a

b

General matrix of dimensions 6×6
with a first order cycle structure of period 2

Figure 13

Matrix

Second order cycle structure

a

b

Figure 14

Figures
13 and
14 display the general matrices of dimensions
6x6 that have a first order or a second order cycle structure of period
2.

Let A and
B be matrices of dimensions (2m) ×
(2m) that have a first order cycle structure of period 2. The product
AB has a second order cycle structure of period 2.

Theorem 4:

Let A and
B be matrices of dimensions (2m) ×
(2m) that have a first order cycle structure of period 2. The matrices
AB and BA are symmetrical to one another: if one reflects
AB about its secondary axis, one obtains
BA.

1

2

3

4

5

6

-1

2

-2

1

3

4

2

4

1

6

3

5

2

1

-1

4

-2

3

3

1

5

2

6

4

-2

-1

3

2

4

1

4

6

2

5

1

3

1

4

2

3

-1

-2

5

3

6

1

4

2

3

-2

4

-1

1

2

6

5

4

3

2

1

4

3

1

-2

2

-1

A B

40

25

39

10

24

9

40

39

25

24

10

9

39

40

24

25

9

10

25

40

10

39

9

24

25

10

40

9

39

24

39

24

40

9

25

10

24

39

9

40

10

25

10

25

9

40

24

39

10

9

25

24

40

39

24

9

39

10

40

25

9

24

10

39

25

40

9

10

24

25

39

40

ABBA

Figure 15

Figure 15 illustrates
an example of the multiplication of two first order cycle matrices of dimensions
6×6 with period 2.

Theorem 5:

Let A be
a matrix of dimensions (2m) × (2m)
that has a first order cycle structure of period 2. A2
has a second order cycle structure of period 1.

Theorem 6:

Let D and
E be matrices of dimensions (2m) ×
(2m) that have a second order cycle structure of period 2. DE
has a second order cycle structure of period 2.

Theorem 7:

Let D and
E be matrices of dimensions (2m) ×
(2m) that have a second order cycle structure of period 2. D
and E commute:
DE=ED.

3

1

4

5

6

2

-1

2

5

3

-2

0

4

3

6

1

2

5

5

-1

-2

2

0

3

1

5

3

2

4

6

2

3

-1

0

5

-2

6

4

2

3

5

1

-2

5

0

-1

3

2

5

2

1

6

3

4

3

0

2

-2

-1

5

2

6

5

4

1

3

0

-2

3

5

2

-1

PQ

18

38

27

4

27

33

27

18

27

38

33

4

38

4

18

33

27

27

27

27

33

18

4

38

4

33

38

27

18

27

33

27

4

27

38

18

PQ
= QP

Figure 16

Figure
16 illustrates an example of the multiplication of two second order
cycle matrices of dimensions 6×6 with cycles
of period 2.

Theorem 8:

Let D be
a matrix of dimensions (2m) × (2m)
that has a first order cycle structure of period 2. Let E be a matrix
of dimensions (2m) × (2m) that
has a second order cycle structure of period 2. Both DE and DE
have a first order cycle structure of period 2.

Figure
17 illustrates an example of the multiplication of a first order and
a second order cycle matrix of dimensions 6×6
with cycles of period 2.

Theorem 9:

Let A be
a matrix of dimensions (2m) × (2m)
that has a first order cycle structure of period 2. The odd powers of A
have a first order cycle structure of period 2, whereas the even powers
of A have a second order cycle structure of period 2.

The following Table
summarises the results obtained with the multiplication of matrices A
and B of dimensions (2m) × (2m)
that have a first or a second order cycle structure of period 2:

This multiplication
table is similar to the multiplication table of negative and positive numbers.
We may call therefore matrices that have a first order cycle structure
negative cycle matrices,
and matrices that have a second order cycle structure
positive
cycle matrices.

Second level generalisation

The cycle structures
discussed so far have period 2. The question arises naturally if other
periods are possible, and if so, what will happen.

As there are 4m
unit squares in each cycle of a (2m) ×
(2m) matrix, any period
p that is a divisor of 4m
may be considered. There exists no problem with the generalisation of first
order cycle matrices. In the case of second order cycle matrices of period
2 we had constant diagonals, that is of period 1. If we consider those
diagonals as degenerated cycles, whereby the upper and the lower parts
of the cycle collapsed, their period of 1 instead of 2 is understood. This
understanding makes it possible to extend the concept of a second order
cycle structure in the following way.

We will say that
a (2m) × (2m) matrix has a second
order cycle structure of period p
if all its second order cycles have period
p, i.e. both the normal
second order cycles and the two diagonals as degenerated cycles have period
p.

2

0

5

6

-4

-1

-1

1

0

2

-5

3

-2

3

1

-3

4

7

2

4

-3

4

2

6

-3

7

3

4

-2

1

4

6

4

2

2

-3

-4

5

-1

2

6

0

-5

0

3

-1

2

1

6

-1

0

-4

2

5

2

3

1

-5

-1

0

4

1

7

-2

-3

3

2

-3

6

2

4

4

EF

Figure 18

-22

23

28

26

12

-7

-30

21

15

9

1

-8

49

7

32

7

36

34

25

15

43

-24

24

51

7

34

7

36

49

32

-24

51

15

24

25

43

12

28

-7

-22

26

23

1

15

-8

-30

9

21

26

-7

23

12

-22

28

9

-8

21

1

-30

15

36

32

34

49

7

7

24

43

51

25

-24

15

EFFE

Figure 19

Figure
18 presents two matrices of dimensions 6×6
that have, in the extended sense, a second order cycle structure of period
4. Their products have also a second order cycle structure of period 4
(see Figure 19). The general form of a second order
cycle matrix of dimensions 6×6 with period 4
is abbaab.

The following generalisations
of theorems 3, 6 and 8 may be proven, consistent with the multiplication
table:

Theorem 3:

Let A and
B be matrices of dimensions (2m) ×
(2m) that have a first order cycle structure of period p.
The product AB has a second order cycle structure of period p.

Theorem 6:

Let A and
B be matrices of dimensions (2m) ×
(2m) that have a second order cycle structure of period p.
The product AB has a second order cycle structure of period p.

Theorem 8:

Let A be
a matrix of dimensions (2m) × (2m)
that has a first order cycle structure of period p. Let B
be a matrix of dimensions (2m) × (2m)
that has a second order cycle structure of period p. Both products
AB and BA have a first order cycle structure of period p.

Third level of generalisation

The dimensions of
the matrices considered so far are (2m) ×
(2m). We may call the matrices therefore even
cycle matrices. We may ask if it is
possible to define odd cycle matrices.

First order cycle structure

Second order cycle structure

Figure 20

The extension of
first order and second order cycle structures to square matrices of odd
dimensions should, if possible, be consistent with the multiplication of
even cycle matrices. It comes out that this is indeed possible. When the
principal diagonal may be considered a collapsed cycle of certain period,
we may call the odd cycle structure a positive or a second order cycle
structure. When the secondary diagonal may be considered a collapsed cycle,
we may call the odd cycle structure a negative or first order cycle structure.
Figure 20 illustrates the first and the second order
cycle structures of dimensions 7×7.

In the case of odd
matrices of dimensions (2m+1) × (2m+1)
the period p has to be a divisor of 2(2m+1).

1

2

-1

-3

4

5

0

3

2

2

1

-2

-1

-2

-1

2

0

1

0

4

3

-3

6

4

1

3

5

0

4

2

3

6

2

-3

0

1

8

5

6

4

2

7

1

0

1

7

5

-3

4

4

5

-4

3

2

7

0

4

1

5

1

8

2

6

5

3

4

2

5

-4

7

4

-3

1

6

5

3

3

0

4

4

-3

-3

6

1

2

0

0

2

4

3

5

3

-3

2

2

0

1

4

-1

-1

0

-1

1

1

2

2

-2

-2

0

4

2

-1

3

1

5

2

-3

2

-2

1

-2

0

2

-1

1

-1

4

3

-3

5

4

1

0

6

3

2

0

3

1

4

-3

0

6

2

1

2

1

6

0

8

7

5

4

-4

4

2

-3

3

7

5

5

4

AB

First
order cycle structure of dimensions 9×9
and period 6

-39

27

5

5

14

25

7

19

26

-27

-7

-17

-22

3

9

-1

4

6

-28

76

9

20

52

71

-1

57

54

54

58

55

30

56

38

48

49

33

63

84

-10

45

68

39

27

34

8

54

55

-4

9

94

53

49

100

72

27

68

8

-10

34

63

39

84

45

49

94

72

-4

100

54

53

55

9

71

57

20

54

76

-1

-28

52

9

38

49

30

33

58

48

54

56

55

7

14

26

5

19

-39

25

27

5

-1

3

6

-17

4

-27

9

-7

-22

25

19

5

26

27

7

-39

14

5

9

4

-22

6

-7

-1

-27

3

-17

-1

52

54

9

57

-28

71

76

20

48

56

33

55

49

54

38

58

30

39

34

45

8

84

27

63

68

-10

53

100

9

72

55

49

54

94

-4

AB
BA

Second
order cycle structure of dimensions 9×9
and period 6

Figure 21

Figure
21 presents some experimentation with the multiplication of 9x9 matrices
that have a first order cycle structure of period 6. The products have
a second order cycle structure of period 6. The reader is invited to do
some further experimentation.

The following theorems
may be proven:

Theorem 3:

If A and
B are matrices of dimensions (2m+1) ×
(2m+1) that have a first order cycle structure of period p, then
AB has a second order cycle structure of period
p.

Theorem 6:

If A and
B are matrices of dimensions (2m+1) ×
(2m+1) that have a second order cycle structure of period p,
then AB also has a second order cycle structure of period p.

Theorem 8:

Let A and
B be matrices of dimensions (2m+1) ×
(2m+1) that have a first order cycle structure and second order
cycle structure of period p, respectively, then
AB and BA
have a first order cycle structure of period p.

Once again, the
same multiplication table holds.

An immediate consequence
of the previous theorems is the following:

Theorem 9:

If A is a
matrix of dimensions (2m+1) × (2m+1)
that has a first order cycle structure of period p, then the even
powers of A have a second order cycle structure of period p
and the odd powers of A have a first order cycle structure of period
p.

Conclusion

Cycle matrices constitute
a generalisation of Liki-matrices. The paper shows the beauty of the reproduction
of both negative and positive cycle structures in the multiplication of
both even and odd cycle matrices of period p. The two cycle structures
may be considered as a new type of symmetry for square matrices.

A

B

AB

First order

First order

Second order

First order

Second order

First order

Second order

First order

First order

Second order

Second order

Second order

General multiplication table of cycle matrices
of period p

Bibliography of books and papers
by Paulus Gerdes related to Lunda-designs.

(1990) On ethnomathematical
research and symmetry, Symmetry: Culture and Science, Budapest,
1(2), 154-170